The principal investigator's proposed research has three parts. The first constructs representations using tools from topology and symplectic geometry, especially the theory of Goresky-Kottwitz-MacPherson, and extends those tools to larger classes of varieties. The second tackles algebraic and combinatorial projects that solve problems in the geometry of the associated varieties. This subject is often called modern Schubert calculus; specific projects include determining multiplication formulas inside the cohomology ring of flag varieties and Grassmannians. The third focuses on computational and enumerative geometric properties of varieties, such as Hessenberg varieties, that arise in areas of representation theory including the Langlands program. The projects will identify fundamental geometric properties of the varieties, like pure-dimensionality, that are critical to advances in representation theory.
Geometric representation theory builds algebraic structures called representations from geometric objects. This leads to deep insights connecting different mathematical disciplines, including algebra, geometry, combinatorics, and mathematical physics. Crucially, a geometric representation establishes a table of correspondences between geometry and algebra. On one hand, the table answers questions in representation theory via geometry; on the other, it answers questions in geometry using algebra and combinatorics. The proposal includes projects to foster student development at all levels and to provide mentoring for women and other underrepresented minorities in mathematics.
